Factoring Trinomials
Sometimes we must use several methods to completely factor a
polynomial.
Procedure â€”
General Strategy for Factoring a Polynomial
Step 1 Factor out the GCF of the terms of the polynomial.
Step 2 Count the number of terms and look for factoring patterns.
Two terms: Look for the difference of two squares, the sum of
two cubes, or the difference of two cubes.
Three terms:
â€¢ Try to factor using the patterns for a perfect
square trinomial.
â€¢ If the trinomial has the form x^{2} + bx + c, try to
find two integers whose product is c and whose
sum is b.
â€¢ If the trinomial has the form ax^{2} + bx + c, try
to find two integers whose product is ac and
whose sum is b.
Four terms: Try factoring by grouping.
Step 3 Factor completely.
To check the factorization, multiply the factors.
Example 1
Factor: 108wx^{2}  36wxy + 3wy^{2}
Solution
Step 1 
Factor out the GCF. Factor each term.
The GCF is 3w.
Factor out 3w. 
108wx^{2} =
2 Â· 2 Â· 3 Â· 3 Â· 3 Â· w Â· x Â· x
= 2 Â· 2 Â· 3 Â· 3 Â· 3 Â· w Â· x
Â· x
= 3w(36x^{2}  12xy + y^{2}) 
 36wxy  2 Â·
2 Â· 3 Â· 3 Â· w Â· x Â· y
 2 Â· 2 Â· 3 Â· 3 Â· w Â· x Â· y

+ 3wy^{2} + 3
Â· w Â· y Â· y
+ 3 Â· w Â· y Â· y 
Step 2 
Count the number of terms and
look for factoring patterns.
There are three terms in 36x^{2}  12xy + y^{2}.
The first term, 36x^{2}, is a perfect square, (6x)^{2}.
The third term, y^{2}, is a perfect square, (y)^{2}.
The middle term, 12xy, is twice the product of the squared terms:
12xy = 2(6x)(y)
This matches the pattern for a perfect square trinomial.
Substitute 6x for a and y for b: a^{2} 2ab b^{2}
`(a b)^{2}
36x^{2}  12xy + y^{2} = (6x)^{2}  2(6x)(y) +
(y)^{2} = (6x  y)^{2} 
Step 3 
Factor completely.
(6x  y)^{2} cannot be factored further. 
Thus, the factorization is 3w(6x  y)^{2}.
You can multiply to check the factorization.
Note:
Donâ€™t forget the GCF, 3w, that we
factored out in Step 1.
Some polynomials cannot be factored using integers.
Example 2
Factor: x^{2} + 3x + 5
Solution
Step 1 
Factor out the GCF.
There are no factors common to all three terms, other than 1 or 1.

Step 2 
Count the number of terms and look for factoring patterns. There are three terms in x^{2}
+ 3x + 5.
This trinomial has the form ax^{2} + bx + c.
To factor this trinomial we must find two integers whose product is
5 and whose sum is 3. Here are the possibilities:
Product 1
Â· 5
1 Â· 5 
Sum 6
6 
There are no integers whose product is 1 and whose sum is 3.
So, the trinomial x^{2} + 3x + 5 cannot be factored using integers.

